legendre_P( n , z )
The Legendre polynomial of z in SageMath. Defined by
\[ P_n (z) = \frac{ 1 }{ 2^n n! } \frac{ d^n }{ dz^n } ( z^2-1 )^n \]A solution of the differential equation
\[ ( 1-z^2 ) \frac{ d^2 f }{ dz^2 } -2z \frac{ d f }{ dz } + n(n+1) f = 0 \]The second linearly independent solution of this equation is legendre_Q.
Equivalent to gen_legendre_P( n , 0 , z ).
Explicit form:
Plot on the real axis:
Special values:
Related functions: legendre_Q gen_legendre_P
Function category: orthogonal polynomials sagemath-docs